Show that $ \int \liminf f_n \leq \liminf \int f_n \leq \limsup \int f_n \leq \int \limsup f_n$

Question

Let $g$ be a non-negative integrable function over $E$ and suppose $\{f_n\}$ is a sequence of measurable functions on $E$ such that for each $n$, $|f_n| \leq g$ a.e. on $E$. Show that $$ \int \liminf f_n \leq \liminf \int f_n \leq \limsup \int f_n \leq \int \limsup f_n.$$

I know that this problem is an application of the Lebesgue dominated convergence theorem.

Any idea of how to go about it thanks, I am really having a hard time with this problem.

Answer 1

By possibly excising a set of measure 0 we can assume that $|f_n| \leq g$ holds on $E$.

Let $$g_n =\inf_{k\geq n} f_k \leq f_n \text{ then } \ g_n \rightarrow \liminf_{n \to \infty} f_n.$$ Note that from $-g \leq |f_n| \leq g$ for all $n$ it also follows that $-g \leq g_n \leq g$ for all $n$ and thus $|g_n| \leq g$.

Using LDCT it follows that $$\int_{E} \liminf_{n \to \infty} f_n = \lim_{n \to \infty} \int_{E} g_n = \liminf_{n \to \infty} \int_{E} g_n \leq \liminf_{n \to \infty} \int_{E} f_n.$$

Also $$\liminf_{n \to \infty}\int_{E} f_n \leq \limsup_{n \to \infty} \int_{E} f_n \tag{$\ast$}.$$

Similarly, let $h_n =\sup_{k \geq n} f_k \geq f_n$ then $h_n \rightarrow \limsup_{n \to \infty} f_n$ and note that $|h_n| \leq g$.

Again, using LDCT we get $$\int_{E} \limsup_{n \to \infty}f_n = \lim_{n \to \infty} \int_{E} h_n = \limsup_{n \to \infty} \int_{E} g_n \geq \limsup_{n \to \infty} \int_{E} f_n\tag{$\ast\ast$}.$$

From above inequalities we have: $$\int_{E} \liminf_{n \to \infty} f_n \leq \liminf_{n \to \infty} \int_{E} f_n \leq \limsup_{n \to \infty} \int_{E} f_n \leq \int_{E} \limsup_{n \to \infty} f_n.$$

Answer 2

Hint: Apply Fatou's Lemma to the sequences $g + f_n, g- f_n.$

Answer 3

We can assume that $|f_n| \leq g$.

let $g_n= $inf $f_k < f_n $. $g_n \rightarrow$ lim inf $f_n$. Since $-g < f_n < g$ for all $n$, it follows that $-g < g_n < g$ thus $|g_n| \leq g$. Now by the lebesgue dominated convergence theorem we get $\int_{E}$ lim inf $f_n =$ lim $\int g_n =$ lim inf $\int_{E} g_n < $ lim inf $\int_{E} f_n$. by the monotonicity of the integral thus we have $\int_{E}$ lim inf $f_n \leq $ lim inf $\int_E f_n \leq$ lim sup $\int_E f_n$